CVPR2010: Sparse Coding and Dictionary Learning for Image Analysis: Part 3: Optimization Techniques for Sparse Coding

Sparse Coding and Dictionary Learning
for Image Analysis
Part III: Optimization for Sparse Coding and Dictionary Learning

Francis Bach, Julien Mairal, Jean Ponce and Guillermo Sapiro

CVPR’10 tutorial, San Francisco, 14th June 2010

Francis Bach, Julien Mairal, Jean Ponce and Guillermo Sapiro Part III: Optimization 1/52

The Sparse Decomposition Problem

1
minp ||x − Dα||2
2 + λψ(α)
α∈R 2
sparsity-inducing
data ﬁtting term regularization

ψ induces sparsity in α. It can be
△
the ℓ0 “pseudo-norm”. ||α||0 = #{i s.t. α[i ] = 0} (NP-hard)
△ p
the ℓ1 norm. ||α||1 = i =1 |α[i ]| (convex)
...
This is a selection problem.


Finding your way in the sparse coding literature. . .
. . . is not easy. The literature is vast, redundant, sometimes
confusing and many papers are claiming victory. . .

The main class of methods are
greedy procedures [Mallat and Zhang, 1993], [Weisberg, 1980]
homotopy [Osborne et al., 2000], [Efron et al., 2004],
[Markowitz, 1956]
soft-thresholding based methods [Fu, 1998], [Daubechies et al.,
2004], [Friedman et al., 2007], [Nesterov, 2007], [Beck and
Teboulle, 2009], . . .
reweighted-ℓ2 methods [Daubechies et al., 2009],. . .
active-set methods [Roth and Fischer, 2008].
...


Matching Pursuit α = (0, 0, 0)
y
d2

d1

z r=x

d3

x

y
d2

d1

z r

d3
< r, d3 > d3
x

y
d2

d1

z r
r− < r, d3 > d3

d3

x

Matching Pursuit α = (0, 0, 0.75)
y
d2

d1

z
r

d3

x

y
d2

d1

z
r

d3

x

Matching Pursuit α = (0, 0.24, 0.75)
y
d2

d1

z

d3
r

x

y
d2

d1

z

d3
r

x

Matching Pursuit

min || x − Dα ||2 s.t. ||α||0 ≤ L
2
α∈Rp
r

1: α←0
2: r ← x (residual).
3: while ||α||0 < L do
4: Select the atom with maximum correlation with the residual

ˆ ← arg max |dT r|
ı i
i =1,...,p

5: Update the residual and the coeﬃcients
T
α[ˆ] ← α[ˆ] + dˆ r
ı ı ı
T
r ← r − (dˆ r)dˆ
ı ı

6: end while

Orthogonal Matching Pursuit α = (0, 0, 0)
y Γ=∅
d2

d1

z x

d3

x

Orthogonal Matching Pursuit α = (0, 0, 0.75)
y Γ = {3}
d2

π2 d1

z x
r

π1
d3
π3

x

Orthogonal Matching Pursuit α = (0, 0.29, 0.63)
y Γ = {3, 2}
d2

d1

z x
π 32 r

π 31 d3

x

Orthogonal Matching Pursuit

min ||x − Dα||2 s.t. ||α||0 ≤ L
2
α∈Rp

1: Γ = ∅.
2: for iter = 1, . . . , L do
3: Select the atom which most reduces the objective

ˆ ← arg min min ||x − DΓ∪{i } α′ ||2
ı ′ 2
i ∈ΓC α

4: Update the active set: Γ ← Γ ∪ {ˆ}.
ı
5: Update the residual (orthogonal projection)
r ← (I − DΓ (DT DΓ )−1 DT )x.
Γ Γ

6: Update the coeﬃcients
αΓ ← (DT DΓ )−1 DT x.
Γ Γ

7: end for

Orthogonal Matching Pursuit
Contrary to MP, an atom can only be selected one time with OMP. It is,
however, more diﬃcult to implement eﬃciently. The keys for a good
implementation in the case of a large number of signals are
Precompute the Gram matrix G = DT D once in for all,
Maintain the computation of DT r for each signal,
Maintain a Cholesky decomposition of (DΓ DΓ )−1 for each signal.
T

The total complexity for decomposing n L-sparse signals of size m with a
dictionary of size p is

O(p 2 m) + O(nL3 ) + O(n(pm + pL2 )) = O(np(m + L2 ))
Gram matrix Cholesky DT r

It is also possible to use the matrix inversion lemma instead of a
Cholesky decomposition (same complexity, but less numerical stability)


Example with the software SPAMS
Software available at http://www.di.ens.fr/willow/SPAMS/

>> I=double(imread(’data/lena.eps’))/255;
>> %extract all patches of I
>> X=im2col(I,[8 8],’sliding’);
>> %load a dictionary of size 64 x 256
>> D=load(’dict.mat’);
>>
>> %set the sparsity parameter L to 10
>> param.L=10;
>> alpha=mexOMP(X,D,param);

On a 8-cores 2.83Ghz machine: 230000 signals processed per second!


Optimality conditions of the Lasso
Nonsmooth optimization

Directional derivatives and subgradients are useful tools for studying
ℓ1 -decomposition problems:
1
min ||x − Dα||2 + λ||α||1
2
α∈Rp 2

In this tutorial, we use the directional derivatives to derive simple
optimality conditions of the Lasso.

For more information on convex analysis and nonsmooth optimization,
see the following books: [Boyd and Vandenberghe, 2004], [Nocedal and
Wright, 2006], [Borwein and Lewis, 2006], [Bonnans et al., 2006],
[Bertsekas, 1999].


Directional derivatives

Directional derivative in the direction u at α:
f (α + tu) − f (α)
∇f (α, u) = lim+
t→0 t
Main idea: in non smooth situations, one may need to look at all
directions u and not simply p independent ones!
Proposition 1: if f is diﬀerentiable in α, ∇f (α, u) = ∇f (α)T u.
Proposition 2: α is optimal iﬀ for all u in Rp , ∇f (α, u) ≥ 0.



1
minp ||x − Dα||2 + λ||α||1
2
α∈R 2
α⋆ is optimal iﬀ for all u in Rp , ∇f (α, u) ≥ 0—that is,

−uT DT (x − Dα⋆ ) + λ sign(α⋆ [i ])u[i ] + λ |ui | ≥ 0,
i ,α⋆ [i ]=0 i ,α⋆ [i ]=0

which is equivalent to the following conditions:

|dT (x − Dα⋆ )| ≤ λ
i if α⋆ [i ] = 0
∀i = 1, . . . , p,
dT (x − Dα⋆ ) = λ sign(α⋆ [i ])
i if α⋆ [i ] = 0


Homotopy
A homotopy method provides a set of solutions indexed by a
parameter.
The regularization path (λ, α⋆ (λ)) for instance!!
It can be useful when the path has some “nice” properties
(piecewise linear, piecewise quadratic).
LARS [Efron et al., 2004] starts from a trivial solution, and follows
the regularization path of the Lasso, which is is piecewise linear.


Homotopy, LARS
[Osborne et al., 2000], [Efron et al., 2004]

|diT (x − Dα⋆ )| ≤ λ if α⋆ [i ] = 0
∀i = 1, . . . , p,
diT (x − Dα⋆ ) = λ sign(α⋆ [i ]) if α⋆ [i ] = 0
(1)
The regularization path is piecewise linear:

DT (x − DΓ α⋆ ) = λ sign(α⋆ )
Γ Γ Γ
α⋆ (λ) = (DT DΓ )−1 (DT x − λ sign(α⋆ )) = A + λB
Γ Γ Γ Γ

A simple interpretation of LARS
Start from the trivial solution (λ = ||DT x||∞ , α⋆ (λ) = 0).
Maintain the computations of |dT (x − Dα⋆ (λ))| for all i .
i
Maintain the computation of the current direction B.
Follow the path by reducing λ until the next kink.


http://www.di.ens.fr/willow/SPAMS/

>> X=normalize(im2col(I,[8 8],’sliding’));
>>
>> %set the sparsity parameter lambda to 0.15
>> param.lambda=0.15;
>> alpha=mexLasso(X,D,param);

Note that it can also solve constrained version of the problem. The
complexity is more or less the same as OMP and uses the same tricks
(Cholesky decomposition).

Coordinate Descent
Coordinate descent + nonsmooth objective: WARNING: not
convergent in general
Here, the problem is equivalent to a convex smooth optimization
problem with separable constraints
1
min ||x − D+ α+ + D− α− ||2 + λα+ 1 + λαT 1 s.t. α− , α+ ≥ 0.
2
T
−
α+ ,α− 2
For this speciﬁc problem, coordinate descent is convergent.
Supposing ||di ||2 = 1, updating the coordinate i :
1
α[i ] ← arg min || x − α[j]dj −βdi ||2 + λ|β|
2
β 2
j=i

r
← sign(di r)(|di r| − λ)+
T T

⇒ soft-thresholding!

http://www.di.ens.fr/willow/SPAMS/

>> X=normalize(im2col(I,[8 8],’sliding’));
>>
>> %set the sparsity parameter lambda to 0.15
>> param.lambda=0.15;
>> param.tol=1e-2;
>> param.itermax=200;
>> alpha=mexCD(X,D,param);



ﬁrst-order/proximal methods

min f (α) + λψ(α)
α∈Rp

f is strictly convex and continuously diﬀerentiable with a Lipshitz
gradient.
Generalize the idea of gradient descent
L
αk+1 ← arg min f (αk )+∇f (αk )T (α − αk )+ ||α − αk ||2 +λψ(α)
2
α∈R 2
1 1 λ
← arg min ||α − (αk − ∇f (αk ))||2 + ψ(α)
2
α∈R 2 L L

When λ = 0, this is equivalent to a classical gradient descent step.


first-order/proximal methods
They require solving efficiently the proximal operator
1 2
minp u−α 2 + λψ(α)
α∈R 2
For the ℓ1 -norm, this amounts to a soft-thresholding:

α⋆ [i ] = sign(u[i ])(u[i ] − λ)+ .

There exists accelerated versions based on Nesterov optimal
first-order method (gradient method with “extrapolation”) [Beck
and Teboulle, 2009, Nesterov, 2007, 1983]
suited for large-scale experiments.


Optimization for Grouped Sparsity
The formulation:
1 2
min x − Dα 2 + λ αg q
α∈Rp 2
g ∈G
data ﬁtting term
group-sparsity-inducing
regularization

The main class of algorithms for solving grouped-sparsity problems are
Greedy approaches
Block-coordinate descent
Proximal methods


Optimization for Grouped Sparsity
The proximal operator:
1 2
minp u−α 2 +λ αg q
α∈R 2
g ∈G

For q = 2,
ug
⋆
αg = ( ug 2 − λ)+ , ∀g ∈ G
ug 2
For q = ∞,
α⋆ = ug − Π
g . 1 ≤λ [ug ], ∀g ∈ G
These formula generalize soft-thrsholding to groups of variables. They
are used in block-coordinate descent and proximal algorithms.


Reweighted ℓ2
Let us start from something simple

a2 − 2ab + b2 ≥ 0.

Then
1 a2
a≤ +b with equality iﬀ a = b
2 b
and
p
1 α[j]2
||α||1 = min + ηj .
ηj ≥0 2 ηj
j=1

The formulation becomes
p
1 2 λ α[j]2
min x − Dα 2 + + ηj .
α,ηj ≥ε 2 2 ηj
j=1


Summary so far
Greedy methods directly address the NP-hard ℓ0 -decomposition
problem.
Homotopy methods can be extremely eﬃcient for small or
medium-sized problems, or when the solution is very sparse.
Coordinate descent provides in general quickly a solution with a
small/medium precision, but gets slower when there is a lot of
correlation in the dictionary.
First order methods are very attractive in the large scale setting.
Other good alternatives exists, active-set, reweighted ℓ2 methods,
stochastic variants, variants of OMP,. . .


Optimization for Dictionary Learning

n
1
min ||xi − Dαi ||2 + λ||αi ||1
2
α∈R p×n 2
D∈C i=1
△
C = {D ∈ Rm×p s.t. ∀j = 1, . . . , p, ||dj ||2 ≤ 1}.

Classical optimization alternates between D and α.
Good results, but very slow!


[Mairal, Bach, Ponce, and Sapiro, 2009a]

Classical formulation of dictionary learning
n
1
min fn (D) = min l (xi , D),
D∈C D∈C n
i =1

where
△ 1
l (x, D) = minp ||x − Dα||2 + λ||α||1 .
2
α∈R 2

Which formulation are we interested in?
n
1
min f (D) = Ex [l (x, D)] ≈ lim l (xi , D)
D∈C n→+∞ n
i =1

[Bottou and Bousquet, 2008]: Online learning can
handle potentially inﬁnite or dynamic datasets,
be dramatically faster than batch algorithms.

Require: D0 ∈ Rm×p (initial dictionary); λ ∈ R
1: A0 = 0, B0 = 0.
2: for t=1,. . . ,T do
3: Draw xt
4: Sparse Coding

1
αt ← arg min ||xt − Dt−1 α||2 + λ||α||1 ,
2
α∈Rp 2

5: Aggregate suﬃcient statistics
At ← At−1 + αt αT , Bt ← Bt−1 + xt αT
t t
6: Dictionary Update (block-coordinate descent)
t
1 1
Dt ← arg min ||xi − Dαi ||2 + λ||αi ||1 .
2
D∈C t 2
i =1

7: end for

Which guarantees do we have?
Under a few reasonable assumptions,
ˆ
we build a surrogate function ft of the expected cost f verifying

ˆ
lim ft (Dt ) − f (Dt ) = 0,
t→+∞

Dt is asymptotically close to a stationary point.

Extensions (all implemented in SPAMS)
non-negative matrix decompositions.
sparse PCA (sparse dictionaries).
fused-lasso regularizations (piecewise constant dictionaries)


Experimental results, batch vs online

m = 8 × 8, p = 256

Experimental results, batch vs online

m = 12 × 12 × 3, p = 512

Inpainting a 12-Mpixel photograph


Extension to NMF and sparse PCA
[Mairal, Bach, Ponce, and Sapiro, 2009b]

NMF extension
n
1
min ||xi − Dαi ||2 s.t. αi ≥ 0,
2 D ≥ 0.
α∈Rp×n 2
D∈C i =1

SPCA extension
n
1
min ||xi − Dαi ||2 + λ||α1 ||1
2
α∈Rp×n 2
D∈C ′ i =1

△
C ′ = {D ∈ Rm×p s.t. ∀j ||dj ||2 + γ||dj ||1 ≤ 1}.
2


Faces: Extended Yale Database B

(a) PCA (b) NNMF (c) DL


Faces: Extended Yale Database B

(d) SPCA, τ = 70% (e) SPCA, τ = 30% (f) SPCA, τ = 10%


Natural Patches

(a) PCA (b) NNMF (c) DL


Natural Patches

(d) SPCA, τ = 70% (e) SPCA, τ = 30% (f) SPCA, τ = 10%


References I
A. Beck and M. Teboulle. A fast iterative shrinkage-thresholding algorithm for linear
inverse problems. SIAM Journal on Imaging Sciences, 2(1):183–202, 2009.
D. P. Bertsekas. Nonlinear programming. Athena Scientiﬁc Belmont, Mass, 1999.
J.F. Bonnans, J.C. Gilbert, C. Lemarechal, and C.A. Sagastizabal. Numerical
optimization: theoretical and practical aspects. Springer-Verlag New York Inc,
2006.
J. M. Borwein and A. S. Lewis. Convex analysis and nonlinear optimization: Theory
and examples. Springer, 2006.
L. Bottou and O. Bousquet. The trade-oﬀs of large scale learning. In J.C. Platt,
D. Koller, Y. Singer, and S. Roweis, editors, Advances in Neural Information
Processing Systems, volume 20, pages 161–168. MIT Press, Cambridge, MA, 2008.
S. P. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press,
2004.
I. Daubechies, M. Defrise, and C. De Mol. An iterative thresholding algorithm for
linear inverse problems with a sparsity constraint. Comm. Pure Appl. Math, 57:
1413–1457, 2004.


References II
I. Daubechies, R. DeVore, M. Fornasier, and S. Gunturk. Iteratively re-weighted least
squares minimization for sparse recovery. Commun. Pure Appl. Math, 2009.
B. Efron, T. Hastie, I. Johnstone, and R. Tibshirani. Least angle regression. Annals of
statistics, 32(2):407–499, 2004.
J. Friedman, T. Hastie, H. H¨lﬂing, and R. Tibshirani. Pathwise coordinate
o
optimization. Annals of statistics, 1(2):302–332, 2007.
W. J. Fu. Penalized regressions: The bridge versus the Lasso. Journal of
computational and graphical statistics, 7:397–416, 1998.
J. Mairal, F. Bach, J. Ponce, and G. Sapiro. Online dictionary learning for sparse
coding. In Proceedings of the International Conference on Machine Learning
(ICML), 2009a.
J. Mairal, F. Bach, J. Ponce, and G. Sapiro. Online learning for matrix factorization
and sparse coding. ArXiv:0908.0050v1, 2009b. submitted.
S. Mallat and Z. Zhang. Matching pursuit in a time-frequency dictionary. IEEE
Transactions on Signal Processing, 41(12):3397–3415, 1993.
H. M. Markowitz. The optimization of a quadratic function subject to linear
constraints. Naval Research Logistics Quarterly, 3:111–133, 1956.


References III
Y. Nesterov. Gradient methods for minimizing composite objective function.
Technical report, CORE, 2007.
Y. Nesterov. A method for solving a convex programming problem with convergence
rate O(1/k 2 ). Soviet Math. Dokl., 27:372–376, 1983.
J. Nocedal and SJ Wright. Numerical Optimization. Springer: New York, 2006. 2nd
Edition.
M. R. Osborne, B. Presnell, and B. A. Turlach. On the Lasso and its dual. Journal of
Computational and Graphical Statistics, 9(2):319–37, 2000.
V. Roth and B. Fischer. The group-lasso for generalized linear models: uniqueness of
solutions and eﬃcient algorithms. In Proceedings of the International Conference
on Machine Learning (ICML), 2008.
S. Weisberg. Applied Linear Regression. Wiley, New York, 1980.


CVPR2010: Sparse Coding and Dictionary Learning for Image Analysis: Part 3: Optimization Techniques for Sparse Coding

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CVPR2010: Sparse Coding and Dictionary Learning for Image Analysis: Part 3: Optimization Techniques for Sparse Coding